BUILT-IN MATHEMATICA SYMBOL

SumConvergence

gives conditions for the sum

to be convergent.

SumConvergence

gives conditions for the multiple sum

to be convergent.

MORE INFORMATION

The following options can be given:

Assumptions	$Assumptions	assumptions to make about parameters
Direction	1	direction of summation
Method	Automatic	method to use for convergence testing

Possible values for Method include:

	"IntegralTest"	the integral test
	"RaabeTest"	Raabe's test
	"RatioTest"	D'Alembert ratio test
	"RootTest"	Cauchy root test

With the default setting Method->Automatic, a number of additional tests specific to different classes of sequences are used.

For multiple sums, convergence tests are performed for each independent variable.

EXAMPLES

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Basic Examples (2)

Test for convergence of the sum

:

Test the convergence of

:

Find the condition for convergence of

:

Test for convergence of the sum

:

In[1]:=

Out[1]=

Test the convergence of

:

In[2]:=

Out[2]=

Find the condition for convergence of

:

In[1]:=

Out[1]=

Scope (14)

Exponential or geometric sums:

Plot the partial sums:

Polynomial exponential sums:

Rational sums:

Convergence picture:

Special functions:

Piecewise functions:

Slowly converging sums in the Abel-Dini scale:

Alternating sums:

Complex-valued sums:

Exponential or geometric series:

Parameter region for convergence:

Power series:

The convergence region for

:

Combined series:

Piecewise sums:

Assuming

to be complex:

A multivariate sum:

Options (4)

The ratio test typically applies to exponential and hypergeometric terms:

In this case the ratio test is inconclusive:

The root test typically applies to exponential terms:

In this case the root test is inconclusive:

The Raabe test works well for rational functions:

In this case the Raabe test is inconclusive:

The integral test works well on logarithmic terms:

In this case the integral test is inconclusive:

Applications (1)

Find the radius of convergence of a power series:

Prove convergence of Ramanujan's formula for

:

Sum it:

Properties & Relations (4)

Convergence properties are not affected by multiplication of constants:

Convergence is not affected by translating arguments:

SumConvergence is automatically called by Sum:

Many conditions generated by Sum are in effect convergence conditions:

With the setting VerifyConvergence->False, typically a regularized value is returned:

SumConvergence is used in sum transforms such as ZTransform:

GeneratingFunction:

ExponentialGeneratingFunction:

FourierSequenceTransform:

Neat Examples (1)

Conditionally convergent periodic sums: